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Lens makers formula

Let's derive the famous lens makers formula. This formula only works for thin lenses. Created by Mahesh Shenoy.

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Video transcript

say we wanted to construct a lens of a particular focal length say 10 centimeters then we might wonder well what does the focal length of a lens really depend on maybe depends on how curved these surfaces are maybe depends on the refractive index of this material so to build this particular lens we need to know the exact relationship between the focal length and these various parameters of that lens and that's exactly what we'll do in this video a lens is just a refracting medium bounded by at least one curved surface in our example we have both the surfaces to be curved and these surfaces are spherical surfaces which means they're both parts of a sphere and a lens is called as a thin lens if the thickness of that lens is much smaller much smaller than the radius of curvature of each of that surface now of course in this diagram the thickness doesn't look a whole that small I've exaggerated the thickness and we've done that because it'll be easier to draw the ray diagrams and show the ray diagrams but we'll assume but we'll just assume that this thickness is actually very smaller than the radius of curvature then we'll call that as a thin lens and our goal now is to figure out what is the focal length of such a thin lens so let's draw our principal axis and O is just the center of this thin lens we usually call it as the optic center and just like for curved mirrors to figure out the focal length we're gonna draw parallel rays of light and see where they meet up one ray of light will draw along the principal axis since this very of light is perpendicular to both the surfaces the angle of incidence is zero there will be no refraction and so this ray will go undeviated how do we know it's perpendicular to the surface well notice that the ray of light is passing through the Centers of curvature and any line through the center of curvature is always perpendicular to the surface so this ray goes under uted the second ray of light will draw above the principal axis now this ray of light will refract it will Bend and to figure out how it bends let's first focus only on the first curved surface so we'll assume the second surface is not there for a while and now to understand how it bends let's draw in normal at this point and that normal is drawn always from the center of curvature and now let's assume that the outside medium is a rarer medium and the inside medium is a denser medium and so the ray of light is moving from rarer to denser the ray of light is going to bend towards the normal so this ray will Bend somewhat like this towards the normal and these two rays are going to meet somewhere over here let's call it point as I that's where the image is formed but of course this is not the final picture because we haven't considered the second surface yet so this is the image if the second surface was not there so now let's consider the second surface now for the second surface this is the incident ray we can totally neglect this initial incident ray and so again if we draw a normal now notice this ray of light is now moving from the denser medium to a rarer medium and so it's going to bend away from the normal so here's our normal so this ray after refraction is going to go away Bend away from the normal and as a result somewhere over here maybe and as a result notice that the initial to paddle rays are being focused after refraction finally at this point so this is our principal focus so here's our principal focus and this distance from here to here is the focal length that's what we have to figure out so how do we figure this out well since we are dealing with refractions at curved surfaces we'll have to use no surprise the curved surface formula so this is how I like to remember the formula our M is the refractive index of the refracted medium meaning it's a medium that contains the refracted ray similarly I M is the refractive index of the incident medium again it means the medium that contains the incident ray and V is the image distance U is the object distance and R is the radius of curvature and since we have two surfaces we'll have to apply this formula to each of this surface so let's start with the first surface when we apply this formula to the first surface we are going to absolutely neglect the second surface that means we are going to neglect this refraction that's happening to the second surface and we assume that the two rays are actually going to meet up at this point and that's the trick to solving problems when we have multiple surfaces we only consider the first surface ignore all the other surfaces all right now let's apply this formula all right the refracted medium is the medium that contains the refracted ray that's over here so it will be this medium let's say it's refractive index is n2 and the incident medium will be the outside medium because this is the incident ray and let's say it's refractive index is n1 so let's substitute that over here so the refracted medium that is n2 divided by V which is the image distance for the first surface this is the image so the image distance is going to be this distance now since our lens is thin whether you consider the distance from this point all considered from this point it's not going to matter so it's the same thing that's the whole reason we're dealing with thin lenses so this is the emails distance V it doesn't matter where you take it from all right so this is the image distance V so it's going to be n2 divided by V minus the incident medium that's over here the outside medium that's n1 + 1 divided by u the object distance where where is the object well there is a flight a paddle and so the object is far away that means infinity so we could say object distance is infinity that's going to be equal to this minus that so that's going to be n 2 minus N 1 minus N 1 divided by R what's R R is the radius of curvature of this surface and since this is the first surface let's call that radius of curvature as R 1 R 1 now you may be curious as to why we're not using sign conventions at all all talk a little bit more about that later on but this is what we got for surface number 1 let's call it as equal now let's apply the equation to the second refracting surface and so for the second or surface after a fraction the image is formed over here and so now we have to apply this formula for the second surface it would be a great idea to pause the video and see if you can try this yourself alright let's do this okay again refracted medium this time notice refraction is happening in the outside medium so this time the refracted medium is N 1 so you'll get N 1 divided by the image distance or our image now is over here think about this for this surface this is the image and the image distance itself is the focal length so divided by the focal length F minus the inside medium well this is the incident ray and therefore now the incident medium is the medium of the lens and so the incident medium is N two divided by the object distance and we might think well the object distance is infinity right well no we have to be very careful in this the important part you see if when you consider these incident rays that's where the objects when the object is at infinity but these were the incident rates for the first surface for our second surface these are the incident Ray's and here's how I like to think about it object is where the incident rays meet simple alright so for example if here was the object then if we drew incident rays like this incident rays like this wherever the incident Ray's meet that would be our object isn't it so we just have to ask ourselves where are these two incident rays meeting well if you backtrack them they're not going to meet anywhere but if you go front they are meeting at this point and it's for that reason it's for that reason this is going to be our object now I know it sounds a little weird to call that as an object because the rays of light are not really meeting at that point they would have met if it wasn't for this surface so we give a name to such kind of an object we call that as a virtual object as if virtual object but you don't have to worry too much about that nevertheless it's the object for our second surface and this is the general way in which how we solve any problem with multiple surfaces the image for the first surface acts like the object for the next surface this is the general way to do problem solving so anyways this is our object for second surface and so our object distance will be V whatever was the image distance before now that same is our object distance all right so this is going to be our object distance and so that will be equal to we have to do n1 minus n2 so n1 minus minus n2 divided by the radius of curvature the radius of curvature of this surface that need not be the same in general for the radius of curvature of the other surface and so we can write as divided by r2 and this now is our equation 2 equation for the second surface and so now if you look at the final picture in reality these rays of light are actually meeting up at this point this was something that we just cooked up assuming what would have happened if this was not there so this V is something that we cooked up and we shouldn't have that in our final equation and so somehow by using these two equations we can get rid of V in fact if you look carefully you have a negative went to by V here you have a positive went to by V here so if we just add the two equations the V cancels out and then we can figure out what the focal length is and so again would be a great idea to pause the video add the two equations and see what you get all right let's do that let's make some space all that is left is now algebra so if we add the two equations these cancel out so on the left hand side we are left with + 1 divided by F minus n1 over infinity what's in 1 over infinity 1 over infinity is 0 so n1 over infinity is also 0 so this gap goes so always left is n1 over F that's equal to we're adding these two right so that'll be N 1 minus n 2 divided by R 2 plus n 2 minus N 1 n 2 minus N 1 / r1 and so finally we can simplify that not finally there might be one more step in wall over there we can take n 2 minus N 1 common so let's do that so if you take n 2 minus N 1 comment out from here we end up with 1 over R 1 over here 1 over R 1 and notice since we took n 2 minus N 1 common there's a negative 1 that remains over here so you get negative 1 1 over R 2 just think about this for a while I just picked out n 2 minus N 1 so a negative 1 remains and so finally we can divide the whole equation by N 1 and so this goes from here and N 1 comes over here and this now we can treat this as our final equation grand equation that uses what the focal length is and if you look at this camera in fact you know what we can make one more simplification if we can divide each term by N 1 and so I just do that over here to save space you will get n 2 by N 1 minus N 1 by N 1 there is just 1 all right so excuse this because that's from the earth space so this is the final equation that tells us what the focal length is and you can see that depends on the refractive indices and the radii of curvature and we give a name to this equation we call this as the lens makers equation or lens makers formula it's called lens makers because tomorrow if you want to make a lens of a suitable focal length then you can use this equation and choose appropriate values of n 2 + 1 R 1 and R 2 so to quickly summarize here the key takeaways of this derivation since we're dealing with curved surfaces for each surface we're going to use the curved surface refraction formula then the image of that first surface is going to be the object for the second surface then you put together equation and we end up with this final result one last question we might have is why we haven't used sign conventions when we substituted in this curved surface formula or think of it this way this is a general formula which works for any case when it comes to curved surface refraction now if we were solving in numerical which deals with specific cases then we would have substituted using science and that's what we always do when we are solving numericals we always substitute science which means whenever you substitute numbers with signs in a general formula you end up solving for that specific case like in a numerical but our goal was not to solve for a specific case our goal was to solve for a general case and that is the reason we didn't use any sign conventions and as a result we ended up with the general formula this will work for any case not just a convex lenss but concave or whatever the thin lens we have